probably assert with safety that between whatever points of temperature we examined ice, there would be found some indication, doubtless almost infinitesimally small, of the apparently abrupt change which was to occur at a higher temperature. It might also be pointed out that all the most important and apparently simple physical laws, such as those of Boyle and Marriotte, Dalton and Gay-Lussac, &c., are only approximately true, and the divergences from observation are forewarnings of abrupt changes, which would otherwise break the law of continuity.

Secondly, it must be remembered that mathematical laws of any complexity will probably present singular cases or negative results, which may present the appearance of discontinuity, as when the law of refraction suddenly yields us with perfect abruptness the entirely different phenomenon of total internal reflection. In the undulatory theory there is no real change of law between the phenomenon of refraction and that of reflection.

Faraday in the earlier part of his career found so many substances possessing more or less magnetic power, that he ventured on a great generalization, and asserted that all bodies shared in the magnetic property of iron. His mistake, as he afterwards himself discovered, consisted in overlooking the fact that though magnetic in a certain sense, some substances might have negative magnetism, and be repelled instead of attracted by the magnet. Between magnetism and diamagnetism there must be a zero near or even at which some substances may be classed, but otherwise magnetic properties appear to be universally present in matter.

Thirdly, where we might expect to find a uniform mathematical law prevailing, the law may undergo abrupt change at singular points, and actual discontinuity may arise. We may sometimes be in danger of treating under

one law phenomena which really belong to different laws. It is generally known, for instance, that a spherical shell of uniform matter attracts an external particle of matter with a force varying inversely as the square of the distance from the centre of the sphere. But this law only holds true so long as the particle is external to the shell. Within the shell the law is wholly different, and the aggregate gravity of the sphere becomes zero, because the force in every direction is neutralized by an exactly equal force. If an infinitely small particle be in the superficies of a sphere, the law is again different, and the attractive power of the shell is half what it would be on particles infinitely close to the surface of the shell. Thus in approaching the centre of a shell from a distance, the force of gravity evinces a double discontinuity in passing through the shell.

It may well admit of question, too, whether discontinuity is really unknown in nature. We perpetually do meet with events which are real breaks upon the previous law, though the discontinuity may then be a sign that some wholly independent cause has come into operation. If the ordinary course of the tides is interrupted by an enormous and irregular wave, we attribute it to an earthquake, or some gigantic natural disturbance. If a meteoric stone falls upon a person and kills him, it is clearly a discontinuity in his life, of which he could have had no anticipation. A sudden sound may pass through the air neither preceded nor followed by any continuous effect. Although, then, we may regard the Law of Continuity as a principle of nature holding rigorously true in of the relations of natural forces, it seems to be a matter of difficulty to assign the limits within which the

many

d Thomson and Tait, Treatise on Natural Philosophy,' vol. i. pp. 346-351.

law is verified. Much caution, therefore, is desirable in its application.

Negative Arguments on the Principle of Continuity.

Upon the principle of continuity we may often found arguments of great force which prove an hypothesis to be impossible, because it would involve a continual repetition of a process ad infinitum, or else a purely arbitrary breach at some point. Bonnet's famous theory of reproduction represented every living creature as containing germs which were perfect representatives of the next generation, so that on the same principle they necessarily included germs of the next generation, and so on indefinitely. The theory was sufficiently refuted when once clearly stated, as in the following poem called the Universee, by Henry Baker:

Each seed includes a plant: that plant, again,
Has other seeds, which other plants contain:
Those other plants have all their seeds, and those
More plants again, successively inclose.

'Thus, ev'ry single berry that we find,
Has, really, in itself whole forests of its kind,
Empire and wealth one acorn may dispense,
By fleets to sail a thousand ages hence.'

The general principle of inference, that what we know of one case must be true of similar cases, if they really are identical in the essential conditions, prevents our asserting anything which we cannot apply time after time under the same circumstances. On this principle Stevinus beautifully demonstrated that weights resting on two inclined planes and balancing each other must be proportional to the lengths of the planes between their apex and a horizontal plane. He imagined an uniform 10), vol. xli. p. 454.

endless chain to be hung over the planes, and to hang below in a symmetrical festoon. If the chain were ever to move by gravity, there would be the same reason for its moving on for ever, and thus producing a perpetual motion. As this is absurd, the portions of the chain lying on the planes, and equal in length to the planes, must balance each other. On similar grounds we may disprove the existence of any self-moving machine, for if it could once alter its own state of motion or rest, in however small a degree, there is no reason why it should not do the like time after time ad infinitum. Even Newton's proof of his third law of motion, in the case of gravity, is of this character. For he remarks that if two gravitating bodies do not exert exactly equal forces in opposite directions, the one exerting the strongest pull will carry both itself and the other away, and will move with constantly increasing velocity ad infinitum. But though the argument might seem sufficiently convincing, Newton in his characteristic way made an experiment with a loadstone and iron floated upon the surface of waterf. In recent years the very foundation of the principle of conservation of energy has been placed on the assumption that it is impossible by any combination whatever of natural bodies to produce force continually from nothings. The principle admits of frequent application in various subtle forms.

Lucretius attempted to prove, by a most ingenious argument of this kind, that matter must be indestructible. For if a definite quantity, however small, were to fall out of existence in any finite time, an equal quantity might be supposed to lapse in every equal interval of time, so that in the infinity of past time the universe must have ceased to existh. But the argument, however ingenious,

f 'Principia,' bk. I. Law iii. Corollary 6.

Helmholtz, Taylor's Scientific Memoirs' (1853), vol. vi. p. 118. hLucretius,' bk. I. lines 232-264.

sems to fail at several points. If past time be infinite, why may not matter have been created infinite also? It would be most reasonable, again, to suppose the matter led in any time to be proportional to the matter

raining, and not to the original quantity; under hypothesis even a finite quantity of original matter od never wholly disappear from the universe. For like reasons we cannot hold that the doctrine of the Conservation of Energy is really proved, or can ever be proved to be absolutely true, however probable it may be regarded on many grounds.

Tendency to Hasty Generalization.

In spite of all the powers and advantages of generalization, men require no incitement to generalize; they are too apt to draw hasty and ill-considered inferences. As Francis Bacon said, our intellects want not wings, but rather weights of lead to moderate their course i. The process is inevitable to the human mind; it begins with childhood and lasts through the second childhood. The child that has once been hurt fears the like result on all Similar occasions, and can with difficulty be made to distish between case and case. It is caution and dis

tion in the adoption of general conclusions that we have to learn, and the whole experience of life is irmed lesson to this effect. Baden Powell has e described this strong natural propensity to nce, and the fondness of the human mind for tances real or fanciful. Our first inducare always imperfect and inconclusive ; wds real evidence by successive approxiwordingly we find false generalization the

mbk. I. Aphorism 104.

Worlds and of Natured edit. p. 16.